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b
−
1
,j
=
0
.
5
b
,
2
j
−
1
+
b
,
2
j
+
0
.
5
b
,
2
j
+
1
,
where
we
set
the
normalization
factors
c
=
0
.
5
for
simplicity.
For
u
=
N
L
j
1
c
L,j
b
L,j
,wehave
=
N
+
1
N
c
+
1
,j
b
+
1
,j
=
c
,j
b
,j
+
d
+
1
,k
ψ
+
1
,k
,
j
=
1
j
=
1
k
∈∇
+
1
and therefore,
c
+
1
,
2
j
+
1
=
0
.
5
c
,j
+
0
.
5
c
,j
+
1
+
d
+
1
,j
+
1
,j
=
2
,...,N
−
1
,
c
+
1
,
2
j
=
c
,j
−
0
.
5
d
+
1
,j
−
0
.
5
d
+
1
,j
+
1
,j
=
2
,...,N
,
(12.2)
c
+
1
,
1
=
0
.
5
c
,
1
+
d
+
1
,
1
,
c
+
1
,N
+
1
=
0
.
5
c
,N
+
d
+
1
,
2
+
1
.
This can be written in matrix form
⎛
⎝
⎛
⎝
⎞
⎠
⎞
⎠
⎛
⎝
⎞
⎠
1
.
5
c
+
1
,
1
c
+
1
,
2
.
.
c
+
1
,N
+
1
−
1
c
+
1
,N
+
1
d
+
1
,
1
c
,
1
.
.
c
,N
d
+
1
,
2
+
1
.
.
.
−
0
.
5
−
0
.
5
.
.
.
0
.
5
0
.
5
=
.
.
.
.
.
.
.
.
.
.
.
.
.
−
0
.
5
0
.
5
0
.
51
−
=
N
L
j
=
Now, starting with
u
1
c
L,j
b
L,j
, we can compute using the decomposi-
tion algorithm (
12.2
) the coefficients
c
L
−
1
,j
and
d
L,k
. Iteratively, we decom-
pose
c
+
1
,j
=
L
=
0
k
∈∇
d
,k
ψ
,k
. Similarly, we can obtain the single-scale coefficients
c
L,j
from the multi-scale wavelet coefficients
d
,k
.
into
c
,j
and
d
+
1
,j
until we have the series representation
u
12.1.2 Norm Equivalences
For preconditioning of the large systems which are solved at each time step, we
require
wavelet norm equivalences
. These are analogous to the classical Parseval
relation in Fourier analysis which allow expressing Sobolev norms of a periodic
function
u
in terms of (weighted) sums of its Fourier coefficients. Wavelets allow for
analogous statements: the Parseval equation is replaced by appropriate inequalities
and the function
u
need not be periodic. Since in the pure jump case
σ
=
0we
H
s
(G)
, it is essential that the wavelet norm
obtain fractional order Sobolev spaces
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